Ok. This is something to work with.
First--

for checking me on my definitions--if I'm going to be consistent anywhere, I better be within my definitions.
As for Webster, his second definition does not indicate religion--there are reasons aside from faith by which we might hold the belief that religion and ecclesiastical affairs should not enter into the functions of the state, esp. into public education. Call it protection from religious persecution, or whatever--beside the point.
In Webster's first definition you provided, we get a choice between "disregards any form of religious faith and worship" and "rejects any form of religious faith and worship." I see that you chose "rejects" which was the stronger case for religion than "disregards." Disregarding religious faith and worship asserts no affirmative (valid/invalid) assertion regarding religious faith and worship, but to use "reject" validly implies an affirmative invalidation of religious faith and worship.
Obviously I'd choose "disregard" because it does not modify definition 2, which is fine by me, and it allows me to accept "rejects" if I'm allowed to reject upon conditions that are not faith-based (to be consistent with my definintions

).
AHA! There it is! I had a problem with Webster's definition of religion because it was self referential and narrow enough that it would exclude schisms, which remain arguably, religions. Likewise, I shall reject (haha!)
"rejects any form of religious faith and worship." since no foundations for rejection are stipulated in the definition, that rejection must assumed to be faith-based (as you pointed out), thus in direct contradiction with itself. Can't have that, because it make the term meaningless.
Just noted: Merriam-Webster<blockquote>
SECULAR:
1 a : of or relating to the worldly or temporal <secular concerns> b : not overtly or specifically religious <secular music> c : not ecclesiastical or clerical <secular courts> <secular landowners>
2 : not bound by monastic vows or rules; specifically : of, relating to, or forming clergy not belonging to a religious order or congregation <a secular priest></blockquote>ALSO<blockquote>
SECULARISM:
: indifference to or rejection or exclusion of religion and religious considerations</blockquote>I'm not attempting to invalidate your point by pulling a different reference that yours, but "secular" as I referenced it does not include this notion of rejection. Secular means not religious, rather than "anti-religious" as the "reject" reading implies. The M-W definition of secularism introduces that "rejection" business, but since it gives me an option, as yours source did (but you didn't dawg!), I'm not willing to choose "reject" on the basis above and that "rejection" is not part of the root definition. Maybe it does in your reference, I don't know.
Now, if I may, address this:If you are not using the word "doctrine" to mean "dogma"--and you could--I won't bother, I have problems with the source's use of "reject" for the reasons stated previous, and particuarly with the assumption inserted: "without any proof." The assumption is fair, I'll concede, but not neccessary if you can accept that something might be rejected for valid reasons (independent of faith, in this instance).
Then I'm afraid you misspoke with:
"... rejects [...] that God does not exist within that worldly spirit or views." It is fair to reject God not existing on the basis that absence of evidence is not proof of absence. I suspect you meant to say:
"... rejects [...] that God does exist within that worldly spirit or views." If so, introducing God--the object of faith--introduces begging the question if you insist that "doctrine" is "dogma" and/or that the rejection stipulated is neccessarily baseless in logical proof or evidence (i.e. faith).
I see it this way:
"secular" is an adjective. Though it can certainly decribe people, it also appropriately describes things. Secular persons are surely non-theistic--without consideration of the existence of God--and may be agnostic--unsure of the existence of God, in neither case is a statement of faith being used, and it certainly is not being rejected, it's just not applicable. This position is so passive that I'm not comfortable calling them "secularists." They're not rejecting the existence of God as an Athieist does, it's more akin to some monumental ignorance of the concept. But those people who advocate, on the basis of evidence, proof of logic, or reason, that some things (not people) should be secular, can fairly be called secularists as far as their advocacy goes. It allows the Pope to say "Render unto Caesar..., render unto the Lord...." The distincion is not that there is a denial of religion, far from it, but that rather religion (theist or not) is not made a consideration in this area or descision.
I don't mind your desire to call a spade a spade ScreamingEagle, but despite being a black suit--clubs are clubs, despite being on cards--diamonds are diamonds, and hearts are hearts. If you think there are people out there who reject any God, you're right, but they are not secularists. Secualrists can believe in God, most of them do. If you think Atheists are trying to wipe out religion, you may be right--but folks of that clan aren't the only ones trying to wipe out all other religions. Secularists practice religion, but it's not secularism--it's Christianity, Judaeism, Islam, Penatcostalism, Lutheranism, Baptism, Catholicism, Hinduism... you name it. Secularists say religion doesn't belong in Governement (for instance) for the same reasons a live grenade doesn't belong in the hands of a four-year-old. Too dangerous. Not a statement of faith. You can still believe in Jesus while believing (for good reasons!) that Hillary Clinton, Ted Kennedy, Barney Frank, Jesse Jackson, Al Sharpton, George Bush, Condoleeza Rice, Pat Robertson, Elvis and/or Jimmy Swaggart should have in no manner any say in how you believe in Jesus--and if you do...WELCOME TO SECULARISM!